For an oriented circle bundle \(\rho : \mathcal {Z} \rightarrow \mathcal {Y}\) , the Chern class is the obstruction to obtaining a closed global 1-cocycle in \(H^1(\mathcal {Z})\)  which restricts to a generator of the cohomology in each fiber. A circle bundle that is not necessarily oriented has a twisted Chern class, which lives in cohomology with coefficients in the orientation presheaf. This class can be defined in terms of the Leray-Serre spectral sequence, and this definition can be extended to other maps with similarly-behaved Leray-Serre spectral sequences, not just bundles. In this section we'll lay out a setup that admits this generalization.
Our discussion is adapted from [1]}.
